classical example of the hybrid system community (https://github.com/dreal/dreal2/tree/master/benchmarks/hybrid_systems/bouncing_ball)
implemented here with reactions and events
%% Cell type:code id: tags:
```
load(ball.bc).
```
%% Cell type:code id: tags:
```
list_model.
```
%%%% Output: execute_result
MA(c)for _=[v]=>x.
g/c for v=>_.
MA(D*c)for v=[v]=>_.
present(x,x0).
present(y,3.5).
parameter(
x0 = 8.725,
D = -0.05,
K = 0.75,
g = 9.8,
c = 1.0
).
add_event(x<=0, c = -K*c).
%% Cell type:code id: tags:
```
list_ode.
```
%%%% Output: execute_result
[0] d(v)/dt= -1*g/c-D*c*v^2
[1] d(x)/dt=c*v
[2] d(y)/dt=0
%% Cell type:markdown id: tags:
# Numerical simulation
In a hybrid system numerical simulations are relatively time consuming due to the necessity of precisely checking the satisfaction time of events
%% Cell type:code id: tags:
```
numerical_simulation. plot.
```
%%%% Output: display_data
%% Cell type:markdown id: tags:
# Model robustness to parameter perturbations
Statistical evaluation by sampling the parameter space
This is time consuming since it involves making one simulation for each parameter set
The FO-LTL formula defines the amplitude range of x in the domain of the free variable h
The FO-LTL formula defines the height of the first bounce of x in the domain of the free variable h
The amplitude 3.96 exceeds here the objective of 3.5, the satisfaction degree is thus greater than one showing some formula robustness
The height 3.96 exceeds here the objective of 3.5, the satisfaction degree is thus greater than one showing some formula robustness
However the model robustness w.r.t. (default) parameter perturbations is below one (0.88) showing that some parameter perturbations destroy the amplitude objective.
However the model robustness w.r.t. (default) parameter perturbations is below one (0.88) showing that some parameter perturbations destroy the height objective.
%% Cell type:code id: tags:
```
%timeout 180
```
%%%% Output: execute_result
Timeout set to 180
%% Cell type:code id: tags:
```
validity_domain(F(x < h /\ F(x > h))).
```
%%%% Output: execute_result
h<3.96646/\h> -1.0e-6
%% Cell type:code id: tags:
```
satisfaction_degree(F(x < h /\ F(x > h)), [h -> 3.5]).
With a satisfaction degree greater than 1 the formula is already satisfied with some margin
Formula robustness can be further optimized by parameter optimization with formula robustness as objective function with no extra cost and much faster computation time than for estimating model robustness
It gives an amplitude of 4.84 which obviously improves the satisfaction degree of the formula
It gives a height of 4.84 which obviously improves the satisfaction degree of the formula
This is shown to also improve the model robustness (0.93) to parameter perturbations as expected in many examples
%% Cell type:code id: tags:
```
seed(0).
search_parameters(F(x < h /\ F(x > h)), [5 <= x0 <= 10, 0.5 <= K <= 1.0 , -0.1 <= D <= 0.0], [h -> 3.5], cmaes_stop_fitness: -0.5).
```
%%%% Output: execute_result
Time: 27.491 s
Stopping reason: Fitness: function value -5.72e-01 <= stopFitness (-5.00e-01)
Best satisfaction degree: 2.339136
[0] parameter(x0=9.555198250463423)
[1] parameter(K=0.8014006370116868)
[2] parameter(D= -0.058468364144163905)
%% Cell type:code id: tags:
```
numerical_simulation. plot.
```
%%%% Output: display_data
%% Cell type:code id: tags:
```
validity_domain(F(x < h /\ F(x > h))).
```
%%%% Output: execute_result
h<4.83916/\h> -1.0e-6
%% Cell type:code id: tags:
```
satisfaction_degree(F(x < h /\ F(x > h)), [h -> 3.5]).
